# Prove that (cosA - cosB) / (sinA + sinB) = ( sinB - sinA ) / ( cosA + cosB )

Question
Prove that $$\displaystyle\frac{{{\cos{{A}}}-{\cos{{B}}}}}{{{\sin{{A}}}+{\sin{{B}}}}}=\frac{{{\sin{{B}}}-{\sin{{A}}}}}{{{\cos{{A}}}+{\cos{{B}}}}}$$

2021-02-16
Cross-multiply:
$$\displaystyle{{\cos}^{{2}}{A}}-{{\cos}^{{2}}{B}}={{\sin}^{{2}}{B}}-{{\sin}^{{2}}{A}}={1}-{{\cos}^{{2}}{B}}-{\left({1}-{{\cos}^{{2}}{A}}\right)}={{\cos}^{{2}}{A}}-{{\cos}^{{2}}{B}}$$,
which is an identity, proving the original equivalence.

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